Integrand size = 15, antiderivative size = 116 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx=-\frac {2 a^5}{b^6 \sqrt {a+\frac {b}{x}}}-\frac {10 a^4 \sqrt {a+\frac {b}{x}}}{b^6}+\frac {20 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^6}-\frac {4 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{b^6}+\frac {10 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^6}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^6} \] Output:
-2*a^5/b^6/(a+b/x)^(1/2)-10*a^4*(a+b/x)^(1/2)/b^6+20/3*a^3*(a+b/x)^(3/2)/b ^6-4*a^2*(a+b/x)^(5/2)/b^6+10/7*a*(a+b/x)^(7/2)/b^6-2/9*(a+b/x)^(9/2)/b^6
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx=-\frac {2 \sqrt {\frac {b+a x}{x}} \left (7 b^5-10 a b^4 x+16 a^2 b^3 x^2-32 a^3 b^2 x^3+128 a^4 b x^4+256 a^5 x^5\right )}{63 b^6 x^4 (b+a x)} \] Input:
Integrate[1/((a + b/x)^(3/2)*x^7),x]
Output:
(-2*Sqrt[(b + a*x)/x]*(7*b^5 - 10*a*b^4*x + 16*a^2*b^3*x^2 - 32*a^3*b^2*x^ 3 + 128*a^4*b*x^4 + 256*a^5*x^5))/(63*b^6*x^4*(b + a*x))
Time = 0.38 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^7 \left (a+\frac {b}{x}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^5}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (-\frac {a^5}{b^5 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 a^4}{b^5 \sqrt {a+\frac {b}{x}}}-\frac {10 \sqrt {a+\frac {b}{x}} a^3}{b^5}+\frac {10 \left (a+\frac {b}{x}\right )^{3/2} a^2}{b^5}-\frac {5 \left (a+\frac {b}{x}\right )^{5/2} a}{b^5}+\frac {\left (a+\frac {b}{x}\right )^{7/2}}{b^5}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^5}{b^6 \sqrt {a+\frac {b}{x}}}-\frac {10 a^4 \sqrt {a+\frac {b}{x}}}{b^6}+\frac {20 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^6}-\frac {4 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{b^6}+\frac {10 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^6}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^6}\) |
Input:
Int[1/((a + b/x)^(3/2)*x^7),x]
Output:
(-2*a^5)/(b^6*Sqrt[a + b/x]) - (10*a^4*Sqrt[a + b/x])/b^6 + (20*a^3*(a + b /x)^(3/2))/(3*b^6) - (4*a^2*(a + b/x)^(5/2))/b^6 + (10*a*(a + b/x)^(7/2))/ (7*b^6) - (2*(a + b/x)^(9/2))/(9*b^6)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65
method | result | size |
orering | \(-\frac {2 \left (256 a^{5} x^{5}+128 a^{4} b \,x^{4}-32 a^{3} b^{2} x^{3}+16 a^{2} b^{3} x^{2}-10 b^{4} x a +7 b^{5}\right ) \left (a x +b \right )}{63 b^{6} x^{6} \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}\) | \(75\) |
gosper | \(-\frac {2 \left (a x +b \right ) \left (256 a^{5} x^{5}+128 a^{4} b \,x^{4}-32 a^{3} b^{2} x^{3}+16 a^{2} b^{3} x^{2}-10 b^{4} x a +7 b^{5}\right )}{63 x^{6} b^{6} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\) | \(77\) |
trager | \(-\frac {2 \left (256 a^{5} x^{5}+128 a^{4} b \,x^{4}-32 a^{3} b^{2} x^{3}+16 a^{2} b^{3} x^{2}-10 b^{4} x a +7 b^{5}\right ) \sqrt {-\frac {-a x -b}{x}}}{63 x^{4} b^{6} \left (a x +b \right )}\) | \(83\) |
risch | \(-\frac {2 \left (a x +b \right ) \left (193 a^{4} x^{4}-65 a^{3} b \,x^{3}+33 a^{2} b^{2} x^{2}-17 a \,b^{3} x +7 b^{4}\right )}{63 b^{6} x^{5} \sqrt {\frac {a x +b}{x}}}-\frac {2 a^{5}}{b^{6} \sqrt {\frac {a x +b}{x}}}\) | \(86\) |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (-126 \sqrt {a \,x^{2}+b x}\, a^{\frac {15}{2}} x^{8}+63 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{7} b \,x^{8}-126 a^{\frac {15}{2}} \sqrt {x \left (a x +b \right )}\, x^{8}-63 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{7} b \,x^{8}+315 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {13}{2}} x^{6}-252 \sqrt {a \,x^{2}+b x}\, a^{\frac {13}{2}} b \,x^{7}+126 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{6} b^{2} x^{7}-63 a^{\frac {13}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} x^{6}-252 a^{\frac {13}{2}} \sqrt {x \left (a x +b \right )}\, b \,x^{7}-126 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{6} b^{2} x^{7}+508 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {11}{2}} b \,x^{5}-126 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} b^{2} x^{6}+63 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b^{3} x^{6}-126 a^{\frac {11}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} x^{6}-63 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b^{3} x^{6}+128 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{2} x^{4}-32 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{3} x^{3}+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4} x^{2}-10 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{5} x +7 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{6}\right )}{63 x^{5} \sqrt {x \left (a x +b \right )}\, b^{7} \left (a x +b \right )^{2} \sqrt {a}}\) | \(541\) |
Input:
int(1/(a+b/x)^(3/2)/x^7,x,method=_RETURNVERBOSE)
Output:
-2/63*(256*a^5*x^5+128*a^4*b*x^4-32*a^3*b^2*x^3+16*a^2*b^3*x^2-10*a*b^4*x+ 7*b^5)/b^6/x^6*(a*x+b)/(a+b/x)^(3/2)
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx=-\frac {2 \, {\left (256 \, a^{5} x^{5} + 128 \, a^{4} b x^{4} - 32 \, a^{3} b^{2} x^{3} + 16 \, a^{2} b^{3} x^{2} - 10 \, a b^{4} x + 7 \, b^{5}\right )} \sqrt {\frac {a x + b}{x}}}{63 \, {\left (a b^{6} x^{5} + b^{7} x^{4}\right )}} \] Input:
integrate(1/(a+b/x)^(3/2)/x^7,x, algorithm="fricas")
Output:
-2/63*(256*a^5*x^5 + 128*a^4*b*x^4 - 32*a^3*b^2*x^3 + 16*a^2*b^3*x^2 - 10* a*b^4*x + 7*b^5)*sqrt((a*x + b)/x)/(a*b^6*x^5 + b^7*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 9534 vs. \(2 (100) = 200\).
Time = 5.87 (sec) , antiderivative size = 9534, normalized size of antiderivative = 82.19 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b/x)**(3/2)/x**7,x)
Output:
-512*a**(47/2)*b**(91/2)*x**19*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/ 2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 2866 5*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a** (29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/ 2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b **60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62* x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/ 2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 7424*a* *(45/2)*b**(93/2)*x**18*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 94 5*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(3 3/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)* b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**5 8*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x* *(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/ 2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 94 5*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 50112*a**(43/2 )*b**(95/2)*x**17*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**( 37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b **54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56* x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x...
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}}}{9 \, b^{6}} + \frac {10 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a}{7 \, b^{6}} - \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{2}}{b^{6}} + \frac {20 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3}}{3 \, b^{6}} - \frac {10 \, \sqrt {a + \frac {b}{x}} a^{4}}{b^{6}} - \frac {2 \, a^{5}}{\sqrt {a + \frac {b}{x}} b^{6}} \] Input:
integrate(1/(a+b/x)^(3/2)/x^7,x, algorithm="maxima")
Output:
-2/9*(a + b/x)^(9/2)/b^6 + 10/7*(a + b/x)^(7/2)*a/b^6 - 4*(a + b/x)^(5/2)* a^2/b^6 + 20/3*(a + b/x)^(3/2)*a^3/b^6 - 10*sqrt(a + b/x)*a^4/b^6 - 2*a^5/ (sqrt(a + b/x)*b^6)
\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} x^{7}} \,d x } \] Input:
integrate(1/(a+b/x)^(3/2)/x^7,x, algorithm="giac")
Output:
integrate(1/((a + b/x)^(3/2)*x^7), x)
Time = 0.67 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx=\frac {34\,a\,\sqrt {a+\frac {b}{x}}}{63\,b^3\,x^3}-\frac {2\,\sqrt {a+\frac {b}{x}}}{9\,b^2\,x^4}-\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {386\,a^4}{63\,b^5}+\frac {512\,a^5\,x}{63\,b^6}\right )}{b+a\,x}-\frac {22\,a^2\,\sqrt {a+\frac {b}{x}}}{21\,b^4\,x^2}+\frac {130\,a^3\,\sqrt {a+\frac {b}{x}}}{63\,b^5\,x} \] Input:
int(1/(x^7*(a + b/x)^(3/2)),x)
Output:
(34*a*(a + b/x)^(1/2))/(63*b^3*x^3) - (2*(a + b/x)^(1/2))/(9*b^2*x^4) - (( a + b/x)^(1/2)*((386*a^4)/(63*b^5) + (512*a^5*x)/(63*b^6)))/(b + a*x) - (2 2*a^2*(a + b/x)^(1/2))/(21*b^4*x^2) + (130*a^3*(a + b/x)^(1/2))/(63*b^5*x)
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx=\frac {\frac {512 \sqrt {a}\, \sqrt {a x +b}\, a^{4} x^{5}}{63}-\frac {512 \sqrt {x}\, a^{5} x^{5}}{63}-\frac {256 \sqrt {x}\, a^{4} b \,x^{4}}{63}+\frac {64 \sqrt {x}\, a^{3} b^{2} x^{3}}{63}-\frac {32 \sqrt {x}\, a^{2} b^{3} x^{2}}{63}+\frac {20 \sqrt {x}\, a \,b^{4} x}{63}-\frac {2 \sqrt {x}\, b^{5}}{9}}{\sqrt {a x +b}\, b^{6} x^{5}} \] Input:
int(1/(a+b/x)^(3/2)/x^7,x)
Output:
(2*(256*sqrt(a)*sqrt(a*x + b)*a**4*x**5 - 256*sqrt(x)*a**5*x**5 - 128*sqrt (x)*a**4*b*x**4 + 32*sqrt(x)*a**3*b**2*x**3 - 16*sqrt(x)*a**2*b**3*x**2 + 10*sqrt(x)*a*b**4*x - 7*sqrt(x)*b**5))/(63*sqrt(a*x + b)*b**6*x**5)